Commensurability and QI classification of free products of finitely generated abelian groups

TL;DR

This paper characterizes groups quasi-isometric to free products of finitely generated abelian groups, showing they are commensurable with a specific free product involving  and ^n groups based on rank data.

Contribution

It provides a classification of groups quasi-isometric to such free products, identifying their commensurability class based on abelian rank data.

Findings

Groups quasi-isometric to free products of finitely generated abelian groups are classified.

Such groups are commensurable with a free product involving  and ^n groups.

The classification depends on the ranks of the abelian factors.

Abstract

Suppose a group $G$ is quasi-isometric to a free product of a finite set $S$ of finitely generated abelian groups; let $S'$ denote the set of ranks of the free abelian parts of the groups in $S$. Then $G$ is commensurable with the free product of $\Z$ with a $\Z^n$ for each $n$ occurring in $S'$.